1.

If x=9-4√5, find the value of x² + 1/x²​

Answer»

QUESTION :–

▪︎ If { \bold{x = 9 - 4 \sqrt{5} }} then find the value of { \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = ? }}

ANSWER :–

\\ \longrightarrow{ \red {\boxed{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = 322 }}}} \\

EXPLANATION :–

GIVEN :–

\\ { \implies \: \bold{x = 9 - 4 \sqrt{5} }} \\

TO FIND :–

\\ \implies { \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = ? }} \\

SOLUTION :–

• First , we have to find \: \: \: { \bold{ \dfrac{1}{x} \: \: - }} \\

\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{1}{9 - 4 \sqrt{5} } }} \\

• Now Rationalization –

\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{1}{9 - 4 \sqrt{5} } \times \frac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} } }} \\

\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{9 + 4 \sqrt{5} }{(9)^{2} - (4 \sqrt{5}) ^{2} } }} \\

\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{9 + 4 \sqrt{5} }{81 - 80 } }} \\

\\ \implies{ \bold{ \dfrac{1}{x} \: \: = 9 + 4 \sqrt{5} }} \\

• We know a property –

\\ \implies{ \pink{ \boxed{\bold{ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab }}}} \\

• So that –

\\ \implies{ \bold{ {(x + \frac{1}{x} )}^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } + 2( \cancel x)( \frac{1}{ \cancel x} ) }} \\

\\ \implies{ \bold{ {(x + \frac{1}{x} )}^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 }} \\

• We should write this as –

\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {(x + \frac{1}{x} )}^{2} - 2 }} \\

• Now PUT the values –

\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {(9 - \cancel{4 \sqrt{5} } +9 + { \cancel{4 \sqrt{5} }} )}^{2} - 2 }} \\

\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {(18)}^{2} - 2 }} \\

\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = 324 - 2 }} \\

\\ \implies{ \boxed{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = 322 }}} \\


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